Approximations Of Stochastic Differential Equations With Time Delay

In this paper, we mainly consider the Euler approximation to stochastic differ-ential equations with time delay under non-Lipschitz conditions, and give the approx-imate order for its L and L2convergence. We elaborate our points from followingchapters.In chapter one, we simply introduce the development and form of stochasticdifferential equations so that the readers can more easily comprehend this paper. Inaddition, we also represent the stochastic process and stochastic integral because ofits basis. Then we give several relative lemmas in the forth subsection, they are quiteuseful in the application of stochastic differential equations.In chapter two, we sketch some relative concepts about the approximation ofstochastic differential equations for our preparation. If we study its approximationsolutions it is necessary on condition that the existence of stochastic differential equa-tions, so we do it. In the second subsection we give the defnition of the convergenceorder of the approximation solutions and introduce some frequently-used methodsabout the approximation solutions of stochastic differential equations in the sequentsubsection.Chapter three is the central of this paper. We mainly study the approximation ofsome stochastic differential equations with time delay under non-Lipschitz conditions.Firstly, we give the defnition of stochastic differential equations with time delay, then,give the form of Euler method to them. Lastly, under suitable assumptions and somelemmas, we proof the frst and second-order convergence of Euler method.Chapter four is also emphasis of our paper. Here, we consider the Taylor approx-imation of some stochastic integro-differential equations.